open I; goal thy "! a m s s' t n. \ \ (new_tv m a & new_tv m s) --> I e a m s = Ok(s',t,n) --> \ \ ( ? r. W e ($ s a) m = Ok(r, $ s' t, n) & s' = ($ r o s) )"; by (expr.induct_tac "e" 1); (* case Var n *) by (simp_tac (!simpset addsimps [app_subst_list] setloop (split_inside_tac [expand_if])) 1); (* case Abs e *) by (asm_full_simp_tac (!simpset setloop (split_inside_tac [expand_bind])) 1); by (strip_tac 1); by (rtac conjI 1); by (strip_tac 1); by (REPEAT (etac allE 1)); by (etac impE 1); by (fast_tac (HOL_cs addss (!simpset addsimps [new_tv_subst])) 2); by (fast_tac (HOL_cs addIs [new_tv_Suc_list RS mp,new_tv_subst_le, less_imp_le,lessI]) 1); (** LEVEL 10 **) by (strip_tac 1); by (REPEAT (etac allE 1)); by (etac impE 1); by (fast_tac (HOL_cs addss (!simpset addsimps [new_tv_subst])) 2); by (fast_tac (HOL_cs addIs [new_tv_Suc_list RS mp,new_tv_subst_le, less_imp_le,lessI]) 1); (** LEVEL 15 **) (* case App e1 e2 *) by (simp_tac (!simpset setloop (split_inside_tac [expand_bind])) 1); by (strip_tac 1); by (rename_tac "s1' t1 n1 s2' t2 n2 sa" 1); by (rtac conjI 1); by (fast_tac (HOL_cs addss !simpset) 1); by (strip_tac 1); by (rename_tac "s1 t1' n1'" 1); by (eres_inst_tac [("x","a")] allE 1); by (eres_inst_tac [("x","m")] allE 1); by (eres_inst_tac [("x","s")] allE 1); by (eres_inst_tac [("x","s1'")] allE 1); by (eres_inst_tac [("x","t1")] allE 1); by (eres_inst_tac [("x","n1")] allE 1); by (eres_inst_tac [("x","a")] allE 1); by (eres_inst_tac [("x","n1")] allE 1); by (eres_inst_tac [("x","s1'")] allE 1); by (eres_inst_tac [("x","s2'")] allE 1); by (eres_inst_tac [("x","t2")] allE 1); by (eres_inst_tac [("x","n2")] allE 1); (** LEVEL 34 **) by (rtac conjI 1); by (strip_tac 1); by (mp_tac 1); by (mp_tac 1); by (etac exE 1); by (etac conjE 1); by (etac impE 1); by ((forward_tac [new_tv_subst_tel] 1) THEN (atac 1)); by ((dres_inst_tac [("a","$ s a")] new_tv_W 1) THEN (atac 1)); by (fast_tac (HOL_cs addDs [sym RS W_var_geD,new_tv_subst_le,new_tv_list_le] addss !simpset) 1); by (fast_tac (HOL_cs addss (!simpset addsimps [subst_comp_tel])) 1); (** LEVEL 45 **) by (strip_tac 1); by (rename_tac "s2 t2' n2'" 1); by (rtac conjI 1); by (strip_tac 1); by (mp_tac 1); by (mp_tac 1); by (etac exE 1); by (etac conjE 1); by (etac impE 1); by ((forward_tac [new_tv_subst_tel] 1) THEN (atac 1)); by ((dres_inst_tac [("a","$ s a")] new_tv_W 1) THEN (atac 1)); by (fast_tac (HOL_cs addDs [sym RS W_var_geD,new_tv_subst_le,new_tv_list_le] addss !simpset) 1); by (fast_tac (HOL_cs addss (!simpset addsimps [subst_comp_tel,subst_comp_te])) 1); by (strip_tac 1); by (mp_tac 1); (** LEVEL 60 **) by (mp_tac 1); by (etac exE 1); by (etac conjE 1); by (etac impE 1); by ((forward_tac [new_tv_subst_tel] 1) THEN (atac 1)); by ((dres_inst_tac [("a","$ s a")] new_tv_W 1) THEN (atac 1)); by (fast_tac (HOL_cs addDs [sym RS W_var_geD,new_tv_subst_le,new_tv_list_le] addss !simpset) 1); by (mp_tac 1); by (REPEAT (eresolve_tac [exE,conjE] 1)); by (REPEAT(EVERY1 [asm_full_simp_tac (!simpset addsimps [subst_comp_tel,subst_comp_te]), REPEAT o etac conjE, hyp_subst_tac])); (** LEVEL 70 **) by (safe_tac HOL_cs); by (simp_tac (!simpset addsimps [o_def,subst_comp_te]) 2); by (subgoal_tac "new_tv n2 s & new_tv n2 r & new_tv n2 ra" 1); by (asm_full_simp_tac (!simpset addsimps [new_tv_subst]) 1); by ((forward_tac [new_tv_subst_tel] 1) THEN (atac 1)); by ((dres_inst_tac [("a","$ s a")] new_tv_W 1) THEN (atac 1)); by (safe_tac HOL_cs); by (best_tac (HOL_cs addDs[sym RS W_var_geD,new_tv_subst_le,new_tv_list_le] addss !simpset) 1); by (fast_tac (HOL_cs addDs [sym RS W_var_geD,new_tv_subst_le,new_tv_list_le] addss !simpset) 1); (** LEVEL 79 **) by (dres_inst_tac [("e","expr1")] (sym RS W_var_geD) 1); by ((dtac new_tv_subst_tel 1) THEN (atac 1)); by ((dres_inst_tac [("ts","$ s a")] new_tv_list_le 1) THEN (atac 1)); by ((dtac new_tv_subst_tel 1) THEN (atac 1)); by (best_tac (HOL_cs addDs [new_tv_W] addss (!simpset addsimps [subst_comp_tel])) 1); (** LEVEL 84 **) qed_spec_mp "I_correct_wrt_W"; (*** We actually want the corollary goal I.thy "I e [] m id_subst = Ok(s,t,n) --> W e [] m = Ok(s, $s t, n)"; by (cut_facts_tac [(read_instantiate[("x","id_subst")] (read_instantiate[("x","[]")](thm RS spec) RS spec RS spec))] 1); by (Full_simp_tac 1); by (fast_tac HOL_cs 1); qed; assuming that thm is the undischarged version of I_correct_wrt_W. Wait until simplification of thms is possible. ***) val lemma = I_correct_wrt_W COMP swap_prems_rl; goal I.thy "!a m s. \ \ new_tv m a & new_tv m s --> I e a m s = Fail --> W e ($s a) m = Fail"; by (expr.induct_tac "e" 1); by (simp_tac (!simpset addsimps [app_subst_list] setloop (split_tac [expand_if])) 1); by (Simp_tac 1); by (strip_tac 1); by (rtac conjI 1); by (strip_tac 1); by (subgoal_tac "TVar m # $ s a = $s(TVar m # a)" 1); by (asm_simp_tac (HOL_ss addsimps [new_tv_Suc_list, lessI RS less_imp_le RS new_tv_subst_le]) 1); by (etac conjE 1); by (dtac (new_tv_not_free_tv RS not_free_impl_id) 1); by (Asm_simp_tac 1); by (strip_tac 1); by (etac exE 1); by (split_all_tac 1); by (Full_simp_tac 1); (** LEVEL 15 **) by (Asm_simp_tac 1); by (strip_tac 1); by (etac exE 1); by (REPEAT(etac conjE 1)); by (split_all_tac 1); by (Full_simp_tac 1); by (dtac lemma 1); by (fast_tac HOL_cs 1); (** LEVEL 23 **) by (etac exE 1); by (etac conjE 1); by (hyp_subst_tac 1); by (Asm_simp_tac 1); by (etac disjE 1); by (rtac disjI1 1); (** LEVEL 29 **) by (full_simp_tac (!simpset addsimps [o_def,subst_comp_tel]) 1); by (EVERY[etac allE 1, etac allE 1, etac allE 1, etac impE 1, etac impE 2, atac 2, atac 2]); by (rtac conjI 1); by (fast_tac (HOL_cs addIs [W_var_ge RS new_tv_list_le]) 1); by (rtac new_tv_subst_comp_2 1); by (fast_tac (HOL_cs addIs [W_var_ge RS new_tv_subst_le]) 1); by (fast_tac (HOL_cs addSIs [new_tv_subst_tel]addIs[new_tv_W RS conjunct1])1); by (rtac disjI2 1); by (etac exE 1); by (split_all_tac 1); by (etac conjE 1); (** LEVEL 40 **) by (Full_simp_tac 1); by (dtac lemma 1); by (rtac conjI 1); by (fast_tac (HOL_cs addIs [W_var_ge RS new_tv_list_le]) 1); by (rtac new_tv_subst_comp_1 1); by (fast_tac (HOL_cs addIs [W_var_ge RS new_tv_subst_le]) 1); by (fast_tac (HOL_cs addSIs [new_tv_subst_tel]addIs[new_tv_W RS conjunct1])1); by (etac exE 1); by (etac conjE 1); by (hyp_subst_tac 1); (** LEVEL 50 **) by (asm_full_simp_tac (!simpset addsimps [o_def,subst_comp_te RS sym,subst_comp_tel RS sym]) 1); qed_spec_mp "I_complete_wrt_W"; (*** We actually want the corollary "I e [] m id_subst = Fail ==> W e [] m = Fail"; Wait until simplification of thms is possible. ***)